Multifractal Nature Research Articles

Long-range temporal correlation in Auditory Brainstem Responses to Spoken Syllable/da/

The speech auditory brainstem response (sABR) is an objective clinical tool to diagnose particular impairments along the auditory brainstem pathways. We explore the scaling behavior of the brainstem in response to synthetic /da/ stimuli using a proposed pipeline including Multifractal Detrended Moving Average Analysis (MFDMA) modified by Singular Value Decomposition. The scaling exponent confirms that all normal sABR are classified into the non-stationary process. The average Hurst exponent is H = 0:77 ± 0:12 at 68% confidence interval indicating long-range correlation which shows the first universality behavior of sABR. Our findings exhibit that fluctuations in the sABR series are dictated by a mechanism associated with long-term memory of the dynamic of the auditory system in the brainstem level. The q-dependency of h(q) demonstrates that underlying data sets have multifractal nature revealing the second universality behavior of the normal sABR samples. Comparing Hurst exponent of original sABR with the results of the corresponding shuffled and surrogate series, we conclude that its multifractality is almost due to the long-range temporal correlations which are devoted to the third universality. Finally, the presence of long-range correlation which is related to the slow timescales in the subcortical level and integration of information in the brainstem network is confirmed.

Multifractal detrended fluctuation analysis on high-frequency SZSE in Chinese stock market

We use multifractal detrended fluctuation analysis (MF-DFA) method to investigate the multifractal behavior of Shenzhen Component Index (SZSE) 5-minute high-frequency stock data from 2017.6.15 - 2018.4.11. We determine generalized Hurst exponent and singularity spectrum and find that these fluctuations have multifractal nature. In order to maintain the long-term memory of the stock , all 9696 data are divided into 6 units. According to the multifractal spectrum, the main parameters of the six units are obtained. Comparing the MF-DFA results for the original SZSE high-frequency time series with those for shuffled series, we conclude that the origin of multifractality is due to both the broadness of probability density function and long-range correlation. The generalized Hurst exponent obviously depend on the order of fluctuation function and change with it. The curve of scaling function clearly departs from a straight line, i.e. it shows clearly nonlinear property, and the multifractal spectrum displays the commonly observed bell shape. This will provide an important and theoretical foundation for researching the forecasting of finance markets.

Multifractal analysis of pore structure of Middle Bakken formation using low temperature N2 adsorption and NMR measurements

Understanding the complexity and heterogeneity of pore structure can improve the comprehensive characterization for complex unconventional reservoirs and better exploit those resources. In this paper, multifractal analysis was conducted on low temperature N2 adsorption and NMR T2 distribution for the pore structure investigation in Middle Bakken formation in Williston Basin, USA. Results show that the averages of specific surface area, total pore volume, the average pore size of five N2 adsorption samples are 3.31 m2/g, 0.008 cm3/g, and 19.57 nm, respectively. Six NMR T2 distributions are bimodal and the signals of left dominant peaks are mainly distributed between 0.3 ms and 10 ms. The T2 logarithmic mean value ranges from 2.18 ms to 2.95 ms with an average value of 2.57 ms. Both N2 adsorption isotherms and T2 distribution demonstrated the multifractal nature of different pore sizes. The key multifractal parameter, singularity strength range (Δα) calculated from N2 adsorption and T2 distribution are respectively distributed in scopes of 0.72–1.54 and 1.63–2.89, indicating that the heterogeneity of pore space from T2 distribution is stronger than that from N2 adsorption. It was observed that the specific surface, micro pore volume, and average pore size are correlated well with Δα. There are no any correlations between Δα and porosity and T2lm, but the cumulative porosity and width of right secondary peaks have an impact on the Δα, which means that the heterogeneity difference between the two methods is attributed by larger pore and micro fractures.

Long Range Correlation in Vegetation Over West Africa From 1982 to 2011

Satellite-derived Normalized Difference Vegetation index (NDVI) data records offer important sources for long term correlation modelling over West Africa. In this study, we assessed long range correlations in half monthly NDVI records over West Africa from 1982 to 2011 using GIMMS NDVI. In our analysis, we assessed (a) the annual and seasonal trends obtained using Ordinary Linear Regression, (b) the detrended lag-1-autocorrelation C(1), (c) the Detrended Fluctuation Analysis (DFA) scaling Hurst exponent h and (d) the Multifractal (MF) characteristics of NDVI. Results show that there exist some patterns or trends in the records that persist over time. The value of C(1) for NDVI was obtained as 0.989 is significant at 95% confidence interval. Consequently, the scaling h values of the Hurst DFA showed that about 37.4, 20.5, 41.7 and 0.5% of the vegetated areas are anti-correlated (h <; 0.5), un-correlated (h = 0.5), correlated (0.5 <; h <; 1) and uncorrelated random walk (h = 1), respectively. The trend analysis from Ordinary Least square Regression (OLR) shows that about 54.3, 0.1 and 45.6% of the vegetated areas are positively, uncorrelated and negatively correlated, respectively. Our findings revealed that the DFA method performed better than OLR and the findings could be useful in identifying areas with improved and degraded vegetation, which cannot be properly captured by the OLR method. Accordingly, the comparison of the MF-DFA results of original data to those of shuffled and surrogate series indicated that the multifractal nature of considered time-series is both from PDF and long-range correlations but arguably, MF due to long range correlation dominates over West Africa. The research is therefore helpful in the formulating crop and environmental management policies that may be used to improve ecosystem management using a long term plan (inter-annual) or short term (inter-seasonal) planning.

Multifractal Dynamic Functional Connectivity in the Resting-State Brain.

Assessing the functional connectivity (FC) of the brain has proven valuable in enhancing our understanding of brain function. Recent developments in the field demonstrated that FC fluctuates even in the resting state, which has not been taken into account by the widely applied static approaches introduced earlier. In a recent study using functional near-infrared spectroscopy (fNIRS) global dynamic functional connectivity (DFC) has also been found to fluctuate according to scale-free i.e., fractal dynamics evidencing the true multifractal (MF) nature of DFC in the human prefrontal cortex. Expanding on these findings, we performed electroencephalography (EEG) measurements in 14 regions over the whole cortex of 24 healthy, young adult subjects in eyes open (EO) and eyes closed (EC) states. We applied dynamic graph theoretical analysis to capture DFC by computing the pairwise time-dependent synchronization between brain regions and subsequently calculating the following dynamic graph topological measures: Density, Clustering Coefficient, and Efficiency. We characterized the dynamic nature of these global network metrics as well as local individual connections in the networks using focus-based multifractal time series analysis in all traditional EEG frequency bands. Global network topological measures were found fluctuating–albeit at different extent–according to true multifractal nature in all frequency bands. Moreover, the monofractal Hurst exponent was found higher during EC than EO in the alpha and beta bands. Individual connections showed a characteristic topology in their fractal properties, with higher autocorrelation owing to short-distance connections–especially those in the frontal and pre-frontal cortex–while long-distance connections linking the occipital to the frontal and pre-frontal areas expressed lower values. The same topology was found with connection-wise multifractality in all but delta band connections, where the very opposite pattern appeared. This resulted in a positive correlation between global autocorrelation and connection-wise multifractality in the higher frequency bands, while a strong anticorrelation in the delta band. The proposed analytical tools allow for capturing the fine details of functional connectivity dynamics that are evidently present in DFC, with the presented results implying that multifractality is indeed an inherent property of both global and local DFC.

Characterisation of the global breakage pattern of maltodextrin agglomerates

Breakage is important in several operations in food industry as well as in quality control of brittle products. Recently, the breakage pattern of maltodextrin agglomerates (a food model) has been described by our research group by using multifractal parameters; however, this behaviour has not elucidated and explained completely such pattern from the onset of breaking to the edge of the breakage profile. Therefore, the aim of this work was to analyse the geometry and symmetry of 10 sections of the breakage pattern of maltodextrin agglomerates by means of digital image analysis (DIA) and fractal analysis. Results showed significant differences in monofractal dimension (MFD) values at 30% and 90% of the breakage pattern length, indicating that two important morphometric changes occurred along the breakage pattern: one section characterised by a MFD value of 1.1 (more irregular morphology) and other section with a MFD value of 0.8 (less irregular morphology). To validate and verify the experimental MFD values, Monte Carlo derived models were fitted to experimental data, obtaining predicted MFD values in agreement to those obtained by DIA. Predicted MDF values were mostly affected (more sensitive) by the inputs to the linear equation as compared to those of a sixth-order equation. Therefore, the onset of the breakage pattern had a different morphology in comparison to the edge of the profile. Also, no significant differences in lacunarity values were observed along the two (according to MFD values) sections of the breakage pattern. This study helps to explaining the multifractal nature of the breakage pattern and provides valuable information towards the understanding of the complex phenomenon of breakage and its onset.

Multifractality of jump diffusion processes

We study the local regularity and multifractal nature of the sample paths of jump diffusion processes, which are solutions to a class of stochastic differential equations with jumps. This article extends the recent work of Barral {\it et al.} who constructed a pure jump monotone Markov process with random multifractal spectrum. The class of processes studied here is much larger and exhibits novel features on the extreme values of the spectrum. This class includes Bass' stable-like processes and non-degenerate stable-driven SDEs.

Multifractal analysis application to the study of fat and its infiltration in Iberian ham: Influence of racial and feeding factors and type of slicing

This paper explores the multifractal features of different commercial designations of Iberian ham (acorn 100% Iberian ham, acorn Iberian ham, feed/pasture Iberian ham and feed Iberian ham). This study has been done by taking as input the fatty infiltration patterns obtained from digital image analysis of ham cuts comparing mechanic and manual slicing. The yielded results show the multifractal nature of fatty connective tissue in Iberian ham, only when knife cutting is applied, confirming the differences between the designations according to their genetics and feeding. Thus, the multifractal parameters presented in this work could be considered as additional information for checking Iberian ham quality by using non-destructive methods based on the combination of image analysis and predictive techniques. Meat industry can take advantage of these methods to evaluate meat products, especially when fat-connective tissue with complex pattern distribution is involved.

The effect of dental LED light‐curing unit photoactivation mode on 3D surface morphology of dental nanocomposites evaluated by two‐dimensional multifractal detrended fluctuation analysis

The objective of this study was to evaluate the effect of two photoactivation modes of dental LED light-curing unit (LCUs) (conventional and "Soft Start" mode) on surface texture parameters of two dental resin-based nanocomposites. LED LCUs were considered as standard light-curing devices in contemporary dental practice. Atomic force microscopy (AFM) was applied to investigate surface morphology on 90 × 90 μm2 scanning area through 2D multifractal detrended fluctuation analysis with computational algorithms basis. In order to compare 3D surface roughness at nanometer scale, singularity spectrum f[α] was used which characterize local scale properties of multifractal nature of samples. The results confirmed that larger spectrum width Δα (Δα = αmax - αmin ) of f(α) is associated with non-uniform surface morphology. Moreover, materials whose polymerization was photoactivated by the "soft start" polymerization mode, showed better quality of the surface microstructure with lower values of AFM surface texture parameters.

Fractional Linear Birth-Death Stochastic Process—An Application of Heun's Differential Equation

The method of Heun's differential equation is demonstrated in studying a fractional linear birth-death process (FLBDP) with long memory described by a master equation. The exact analytic solution of the generating function for the probability density is obtained on the basis of Heun's differential equation. The multi-fractal nature of FLBDP associated with long memory is demonstrated in conjunction with the present simple birth death process. Finally, the subtle multi-fractal nature of critical fluctuations under long memory is also displayed in the present FLBDP. Further, discussions are also given on the features of transient fluctuation in systems with long memory.

Blast fragmentation classification using Number-Size (N-S) fractal model in Jalal-Abad iron mine

One of the main purpose of accurate blasting in open pit mining is to achieve optimum rock fragmentation. The degree of rock fragmentation plays a significant role in order to control and minimise the overall production cost including loading, hauling and crushing. In the present paper, the application of a Number-Size (N-S) fractal model is intended to classify the blast fragmentation size in the Jalal-Abad iron mine, SW Iran, using GoldSize image analysis software for four blasting with the obtained result being compared with Kuz-Ram curves. To do this, the fractal dimensions via N-S log-log plots were generated based on the output of the GoldSize software. The results indicated that the fragmented rocks have a multifractal nature with four/five different fragmented populations in terms of size namely; the fine rocks with the size of less than 16 cm, Mean-fragment values between 16 and 45 cm, In-range between 45 and 70 cm and finally, oversize larger than 70 cm.

Identification of defective two dimensional semiconductors by multifractal analysis: The single-layer [formula omitted] case study

Two dimensional semiconductor such as single-layer transition metal dichalcogenides (SL-TMD) have attracted most attentions as an atomically thin layer semiconductor materials. Typically, lattice point defects (sulfur vacancy) created by physical/chemical method during growth stages, have disadvantages on electronic properties. However, photoluminescence (PL) spectroscopy is conventionally used to characterize single-layer films but until now it has not been used to show the presence of defects or estimate their population due to overall similarity of general feature PL spectra. To find a feasible and robust method to determine the presence of point defects on single layer MoS2 without changing the experimental setup, Multifractal Detrended Fluctuation Analysis (MF-DFA) and Multifractal Detrended Moving Average Analysis (MF-DMA) are applied on the PL spectrum of single layer MoS2. We compare the scaling behavior of PL spectrum of pristine and defective single layer MoS2 determined by MF-DFA and MF-DMA. Our results reveal that PL spectrum has multifractal nature and different various population of point defects (sulfur vacancy) on single layer MoS2 change dramatically multifractality characteristics (Hurst, Hölder exponents) of photoluminescence spectrum. It is exhibited creating more lattice point leads to smaller fluctuations in luminescent light that it can help to design special defect structure for light emitted devices. The relative populations of point defects are almost elucidated without utilizing expensive characterization instruments such as scanning tunneling microscopy (STM) and high resolution transmission electron microscopy (HR-TEM).

Multifractality signatures in quasars time series – I. 3C 273

The presence of multifractality in a time series shows different correlations for different time scales as well as intermittent behaviour that cannot be captured by a single scaling exponent. The identification of a multifractal nature allows for a characterization of the dynamics and of the intermittency of the fluctuations in non-linear and complex systems. In this study, we search for a possible multifractal structure (multifractality signature) of the flux variability in the quasar 3C 273 time series for all electromagnetic wavebands at different observation points, and the origins for the observed multifractality. This study is intended to highlight how the scaling behaves across the different bands of the selected candidate which can be used as an additional new technique to group quasars based on the fractal signature observed in their time series and determine whether quasars are non-linear physical systems or not. The Multifractal Detrended Moving Average algorithm (MFDMA) has been used to study the scaling in non-linear, complex and dynamic systems. To achieve this goal, we applied the backward ({\theta} = 0) MFDMA method for one-dimensional signals. We observe weak multifractal (close to monofractal) behaviour in some of the time series of our candidate except in the mm, UV and X-ray bands. The non-linear temporal correlation is the main source of the observed multifractality in the time series whereas the heaviness of the distribution contributes less.

Multifractal Desynchronization of the Cardiac Excitable Cell Network During Atrial Fibrillation. I. Multifractal Analysis of Clinical Data.

Atrial fibrillation (AF) is a cardiac arrhythmia characterized by rapid and irregular atrial electrical activity with a high clinical impact on stroke incidence. Best available therapeutic strategies combine pharmacological and surgical means. But when successful, they do not always prevent long-term relapses. Initial success becomes all the more tricky to achieve as the arrhythmia maintains itself and the pathology evolves into sustained or chronic AF. This raises the open crucial issue of deciphering the mechanisms that govern the onset of AF as well as its perpetuation. In this study, we develop a wavelet-based multi-scale strategy to analyze the electrical activity of human hearts recorded by catheter electrodes, positioned in the coronary sinus (CS), during episodes of AF. We compute the so-called multifractal spectra using two variants of the wavelet transform modulus maxima method, the moment (partition function) method and the magnitude cumulant method. Application of these methods to long time series recorded in a patient with chronic AF provides quantitative evidence of the multifractal intermittent nature of the electric energy of passing cardiac impulses at low frequencies, i.e., for times (≳0.5 s) longer than the mean interbeat (≃ 10−1 s). We also report the results of a two-point magnitude correlation analysis which infers the absence of a multiplicative time-scale structure underlying multifractal scaling. The electric energy dynamics looks like a “multifractal white noise” with quadratic (log-normal) multifractal spectra. These observations challenge concepts of functional reentrant circuits in mechanistic theories of AF, still leaving open the role of the autonomic nervous system (ANS). A transition is indeed observed in the computed multifractal spectra which group according to two distinct areas, consistently with the anatomical substrate binding to the CS, namely the left atrial posterior wall, and the ligament of Marshall which is innervated by the ANS. In a companion paper (II. Modeling), we propose a mathematical model of a denervated heart where the kinetics of gap junction conductance alone induces a desynchronization of the myocardial excitable cells, accounting for the multifractal spectra found experimentally in the left atrial posterior wall area.

Multifractal detrended cross-correlation analysis on air pollutants of University of Hyderabad Campus, India

In this paper, we study the multifractal characteristics and cross-correlation behaviour of Air Pollution Index (API) time series data through multifractal detrended cross-correlation analysis method. We analyse the daily API records of nine air pollutants of the university of Hyderabad campus for a period of three years (2013–2016). The cross-correlation behaviour has been measured from the Hurst scaling exponents and the singularity spectrum quantitatively. From the results, it is found that the cross-correlation analysis shows anti-correlation behaviour for all possible 36 bivariate time series. We also observe the existence of multifractal nature in all the bivariate time series in which many of them show strong multifractal behaviour. In particular, the hazardous particulate matter PM2.5 and inhalable particulate matter PM10 shows anti-correlated behaviour with all air pollutants.

Multifractal detrended cross-correlation analysis on NO, NO[formula omitted] and O[formula omitted] concentrations at traffic sites

NOX plays the important role for O3 production in atmospheric photochemical processes. In this paper, the cross-correlations between NO (NO2) and O3 at three traffic sites in Hong Kong are investigated, using the multifractal detrended cross-correlation analysis (MFDCCA). The results show that the cross-correlations between NO (NO2) and O3 have multifractal nature and long term persistent power-law decaying behavior. The sources of multifractality are discussed based on the shuffling and phase randomization procedure. The chi square test is applied to identify the contributions degree of NO and NO2 to multifractality due to its own long term correlations respectively. And the temporal evolutions of the local contributions degree of NO and NO2 to multifractality are investigated by the sliding windows method. The differences between them are explained by the self-organized criticality mechanism of air pollution, combined with global solar radiation. MFDCCA provides a helpful approach for understanding the quantitative relationship between the O3 and its precursors.

Short-term wind power prediction based on improved small-world neural network

In a competitive electricity market, wind power prediction is important for market participants. However, the prediction has not a general solution due to its inherent uncertainty, intermittency, and multi-fractal nature. This paper firstly constructs a small-world BP neural network (SWBP) with weight convergence and statistics analysis in order to build a maximum approximation for its nonlinear computation. Then, a modified mutual information (MI) is presented to select the input features for the SWBP, whose selection criteria is to establish the relationship between the numerous candidate features of the input and output associated with the wind power prediction by eliminating the redundant. Thirdly, the improved SWBP based on the modified MI is compared with the BP network upon the 15-min-ahead wind power prediction for performance testing, which includes convergence, training time, and forecast accuracy. Moreover, mean value method is adopted to smooth the volatility of selected input. At last, illustrative examples based on the 4-h-ahead rolling prediction are given to demonstrate its stability, validity, and accuracy of the proposed methodology contrasted with the BP, PSOBP, and RBF neural network algorithms.

Multifractal detrended cross correlation analysis of atmospheric CO&lt;SUB align="right"&gt;2 concentration and land-surface temperature anomalies

This paper studies the cross correlation between the time series of atmospheric CO2 concentration and the temperature anomalies data using multifractal detrended cross correlation analysis (MF-DXA) methodology. The data sets for both the series are obtained over a period from 1994 to 2009. In our previous paper we have studied the multifractal nature of the time series of atmospheric CO2 concentration using multifractal detrended fluctuation analysis (MF-DFA). In this paper we have attempted to explore the correlation between the increasing CO2 concentration and increasing values of temperature anomalies. The present study reveals the fact that the individual series are autocorrelated and there exists cross correlation between the two mentioned time series.

Multifractal detrended cross correlation analysis of atmospheric CO&lt;SUB align="right"&gt;2 concentration and land-surface temperature anomalies

This paper studies the cross correlation between the time series of atmospheric CO2 concentration and the temperature anomalies data using multifractal detrended cross correlation analysis (MF-DXA) methodology. The data sets for both the series are obtained over a period from 1994 to 2009. In our previous paper we have studied the multifractal nature of the time series of atmospheric CO2 concentration using multifractal detrended fluctuation analysis (MF-DFA). In this paper we have attempted to explore the correlation between the increasing CO2 concentration and increasing values of temperature anomalies. The present study reveals the fact that the individual series are autocorrelated and there exists cross correlation between the two mentioned time series.

MULTIFRACTAL CROSS WAVELET ANALYSIS

Complex systems are composed of mutually interacting components and the output values of these components usually exhibit long-range cross-correlations. Using wavelet analysis, we propose a method of characterizing the joint multifractal nature of these long-range cross correlations, a method we call multifractal cross wavelet analysis (MFXWT). We assess the performance of the MFXWT method by performing extensive numerical experiments on the dual binomial measures with multifractal cross correlations and the bivariate fractional Brownian motions (bFBMs) with monofractal cross correlations. For binomial multifractal measures, we find the empirical joint multifractality of MFXWT to be in approximate agreement with the theoretical formula. For bFBMs, MFXWT may provide spurious multifractality because of the wide spanning range of the multifractal spectrum. We also apply the MFXWT method to stock market indices, and in pairs of index returns and volatilities we find an intriguing joint multifractal behavior. The tests on surrogate series also reveal that the cross correlation behavior, particularly the cross correlation with zero lag, is the main origin of cross multifractality.